modelling of corona discharge

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 35, NO. 2, MARCH/APRIL 1999

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Modeling of DC Corona Discharge Along an Electrically Conductive Flat Plate with Gas FlowGerald M. Colver and Samir El-KhabiryAbstractThe development of a corona discharge was evaluated numerically over a nite region of a semi-innite at plate having small (ohmic) surface conductivity with owing gas. The model simulates a positive ion corona discharge (ionic wind) in the direction of gas ow generated by two parallel wires mounted ush with the surface of the plate. The deposition and removal of ions at the surface are permitted. Five coupled partial differential equations govern the gas phase model together with empirical equations for electrical discharge (8I characteristic). Two voltage bias case studies were considered: rst, the two electrodes have the same potential but are of opposite sign; and second, the positive electrode carries the full potential with the remaining electrode grounded. Several interesting effects are noted relating to the voltage and current distribution, surface potential, and free-stream velocity. Boundary layer development and surface shear are also discussed. Index Terms Corona discharge, electrically conductive gas, at plate, ionic wind.

I. INTRODUCTION ASEOUS discharges involving corona and ionic wind effects have been widely studied both experimentally and theoretically in a stagnant gas [1][4]. In the case of corona interaction with a surface, several engineering applications have been investigated including surface cooling, electrostatic precipitation, and the effect on ame kinetics [5][11]. With a owing gas, coronas have been shown to inuence boundary layer ow and surface drag [12][16]. Experimental measurements have conrmed a drag effect on a glass plate [16], while in the case of rf discharge (16 kHz) a substantial drag reduction between parallel electrodes was noted [17]. A recent study of corona drag on a at plate suggests that the possible mechanisms of ion deposition or removal from the surface will inuence the drag and thinning of the boundary layer [18], [19]. The present study relates to the aforementioned applications of surface cooling, electrostatic precipitation, andPaper MSDAD-S 9816, presented at the 1997 Industry Applications Society Annual Meeting, New Orleans, LA, October 59, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electrostatic Processes Committee of the IEEE Industry Applications Society. Manuscript released for publication July 10, 1998. G. M. Colver is with the Department of Mechanical Engineering, Iowa State University, Ames, IA 50011 USA (e-mail: [email protected]). S. El-Khabiry was with the Department of Mechanical Engineering, Iowa State University, Ames, IA 50011 USA. He is now with Bergstrom Inc., Rockford, IL 61125 USA (e-mail: [email protected]). Publisher Item Identier S 0093-9994(99)01136-6.

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drag reduction wherein laminar boundary layer interactions (thickness, velocity prole, etc.) with a corona are of interest. For example, boundary layer control can be utilized to affect pressure loss or reduce separation of ow from the walls in low-speed wind tunnels or in owpast air foils [11][14], [16][19]. In surface cooling, the boundary layer thickness plays an important role in controlling heat transfer [7], [15]. Similarly, boundary layers and back corona can possibly inuence reentrainment and the transfer of particulate matter at the walls in electrostatic precipitators [9, pp. 319 and 343]. The interaction of a positive corona with a semi-insulating at plate is investigated for laminar boundary layer ow (Reynolds number 50 000 based on distance along the surface from the leading edge). An atmospheric positive dc corona discharge is driven with the ow by parallel wire electrodes. The boundary layer begins at the leading edge of the plate [20] with the electrodes located just downstream. The electrodes are considered to be mounted ush on the surface so as not to disturb the ow. The upstream-most electrode is assumed to produce only positive ions. The momentum generated by the discharge is coupled with the neutral gas molecules through collisions and randomized ionic motion [1], [10]. The positive corona generates an ionic wind that can either increase or decrease the thickness of the existing boundary layer depending on the polarity of the attached electrodes [21]. For a thinning boundary layer, the ions constituting the space charge transmit a reaction force to the electrodes that leads to a net drag reduction on the plate [18]. A special consideration of the model is that positive ions can be freely exchanged with the conducting surface so that ion deposition and removal are possible without regard to mechanisms other than charge conservation and the electrical conductivity relationships on the surface and in the gas phase. Two voltage bias case studies are considered: rst, the two electrodes have the same potential but are of opposite sign with the adjacent surfaces to the electrodes grounded; second, the positive electrode carries the entire potential with the remaining electrode grounded and the adjacent surfaces are at the potential of the corresponding electrodes. These two possibilities inuence the exchange of current between the electrodes and the surroundings. The development of the corona discharge is investigated as regards the gas and surface potentials, and the current and charge density distributions. An important aspect of the study is the inuence of boundary layer convection on the development of the corona.

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Fig. 1. Flow past ohmic conducting (surface) at plate with corona thinning of the boundary layer from positive ion ow (case study II potentials).

II. MODELING Fig. 1 shows the two-dimensional model of a surface conductive (ohmic) at plate with laminar boundary layer growth starting at the leading edge. Boundary layer thinning takes place over the nite region of accelerating ow adjacent to the electrodes from a positive ion dc discharge. The wire electrodes are assumed to be nite, but of small diameter in comparison to the boundary layer thickness and mounted ush with the surface so as not to disturb the ow. Further details of the corona model are given in [18]. The model utilizes the simultaneous solution of ve coupled partial differential equations governing the gas phase and solid interface as follows: conservation of charge in the gas phase and at the solid interface, Poissons equation of electrostatics, conservation of momentum, and bulk gas continuity. An additional conservation equation in the gas phase satises the total current produced at the positive electrode. The momentum equations were simplied using the thin boundary layer approximation [20] and by neglecting gravitational forces. momentum equation (with the This means that only the ow) is needed to describe the ow at a distance from the leading edge where the Reynolds number (based on distance downstream of the leading edge) is sufciently large ( 1000). In the immediate vicinity of the electrodes where radial elds are large, our calculations show that, for distances from the electrodes greater than about 5% of the electrode gap length, the thin boundary layer approximation is valid [18]. The voltagecurrent ( ) characteristic of the corona itself was approximated using the current model of Seaver [22] t to the data of Soetomo [16] and Peeks formula [1], [23] for the eld breakdown. The simpler case of a wire-to-plane discharge has been modeled in detail [4]. In the present study, the complications of modeling surface electrodes were best handled by a semi-empirical approach. The use of Peeks formula is admittedly articial; however, a precise value of the onset voltage is not critical to the development of our model. A nite wire diameter was utilized only for purposes of estimating the corona onset voltage and current magnitude [24]. A line charge replaced the wire for the eld calculations as a reasonable approximation when the wire thickness is small compared to the boundary layer thickness.

The voltage drop through the corona sheath and its thickness were not considered except through the use of Soetomos data. Realistically, neglecting the ion sheath would be true only for larger wire diameters with the corona sheath determined by gas kinetics and the mean free path [1]. The plate surface was taken to be ohmic with a uniform surface conductivity S-sq. This magnitude of electrical conductivity falls within the range of experimental conductivities for the adsorption of water on a glass surface. Many insulators (glass, y ash, dielectrics, ceramics, etc.) exhibit ohmic surface and bulk electrical conductivities for adsorbed ionic molecules [25], [26]. However, nonohmic behavior in bulk solids is exhibited for large elds ( 10 V/m), such as, for example, in glass and for gas discharge over surfaces [27][30]. Upstream of the leading edge, the ow is corona free, so that only the Laplace equation is utilized for the electrostatics. The remaining boundaries at innity are at zero potential. The gas ow is laminar and parallel to the plate with no slip assumed at the surface. The ow has a constant free-stream velocity (Fig. 1). Downstream of the plate ( direction) the boundary layer grows without limit (the numerical solution domains are nite). The system pressure is taken to be uniform at one atmosphere and normal temperature, so that the gas is incompressible. Only positive ions and neutral molecules are assumed to be present in the gas (air) phase. The ion velocity is evaluated by superposition of the bulk gas ow m /Vs). A and a constant ion mobility ( constant ion mobility is valid if the eld strength is not too large [10], [31]. Ion diffusion was neglected. Unipolar ions of positive polarity can be generated from parallel wire electrodes if the potential difference is just above that for corona onset [1]. In practice, an alternative and more efcient method for generating a positive corona would be to increase the diameter of the downstream electrode. III. FORMULATIONS Detailed descriptions of the governing equations are given in [18]. A summary of the formulations is given here. The steadystate two-dimensional thin boundary layer equations [20] are employed for continuity and momentum in directions (along the ow) and (perpendicular to the ow) with an electric eld body force term added. A. Fluid Equations Continuity is given by (1) and -momentum [20] is (2) where and are the velocity components in the and directions, respectively, ( 1.20 kg/m is the uid density, m /s, dynamic viscosity) is the


COLVER AND EL-KHABIRY: MODELING OF DC CORONA DISCHARGE

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uid kinematic viscosity, is the electric eld strength in is the positive ion charge density. the direction, and Fluid surface and boundary conditions are (3) B. Electrostatic Equations Irrotation of electric eld [32, pp. 88, 92] is given by (4) Poissons equation (Gauss law plus irrotation) [32, p. 96] is as follows: (5) Conservation of charge [18] is given by (6) is the gas phase electric eld strength, is the where is the positive ion mobility, and is electric potential, the permittivity of free space. Peeks formula for electrode breakdown [1], [23] is m Breakdown voltage [1] is kV Current source [18], [22] is (9) A/m Conservation of current at positive electrode is A/m Ionic motion (conduction plus convection) is A/m (12) (11) (10) (8) kV/cm (7)

C. Electrostatic Boundary Conditions at Conservation of charge at the interface (between electrodes and ) [18] is given by (13) Ohms law between the electrodes (surface and gas phase) is A/m and A/m (14)

where the surface conductivity bulk gas conductivity is

is taken as constant and the

Ohms law for surfaces outside the electrodes in case study I; case study II, ( ) is

(15) Laplaces equation solved upstream of the leading edge [18] for case study II is

(16) where is a characteristic distance ( height of computational domain). Remaining gas phase boundaries (both case studies) are (17) at Without ion deposition/removal one has and so recovers from (13) the two surface boundary conditions between the electrodes according to and . The former equation gives the expected . The linear potential drop along the surface as second equation infers a zero (gas phase) current normal to the surface, i.e., without deposition or removal of charge. Equations (5) and (6) together with the boundary condition equations (13) and (14) assure that the potential and electric eld tangent are continuous across the surface interface. The normal components of the eld strength across an interface can be evaluated ex post facto with Gauss continuity condition to reveal the surface charge density [32, pp. 253254]. IV. NUMERICAL METHOD Simple at-plate boundary layers have been investigated using various numerical approaches [33], [34]. The present formulations require the simultaneous solutions of two marching problems, one from (1) and (2) (parabolic when combined), and one from (6) (parabolic or hyperbolic) and an equilibrium problem governed by the elliptic (5). Finite differencing was utilized. The four equations constitute a coupled system with the four unknowns , , , and . The electrode current (11) was integrated using Gauss quadrature. Two outer congruent

are the current densities for the gas phase and where and and ( 0.680 kV for Soetomos surface, respectively, data) are the onset eld strength and voltage, respectively, is unit length of electrode, is the radius of the corona wire is an irregularity factor ( 0.72) for normal in centimeters, wires (unpolished), is the distance between the electrodes, ( 1.0 ) is a factor relating to the gas, ( 76 cm Hg) is the air pressure, ( 298K) is the absolute temperature, is the is the source corona current per length of electrode wire, potential difference between the two electrodes in kV, is the ion charge, is the Boltzmann constant, is a ratio known as the ion-to-neutral excess momentum concentration factor, and is a constant representing the gas-phase resistance outside . the corona wire with units


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computational grids were dened for the solution domain, one for the electrostatics problem and one for the uids problem. The computational domain was discretized with two separate clustered grids in the direction and a uniform grid in the direction. The direction utilized a coordinate transformation for grid clustering and a stretching parameter for each grid. at the In the solution, the value of the charge density positive electrode was adjusted in (12) to satisfy (11). The charge density was assigned zero values on all gas phase boundaries of the computational domain. An interpolation scheme utilizing a cubic spline function was used to transfer the solution velocity components from the uids grid to the electrostatic grid. When a solution on the electrostatic grid was converged, the body force term was interpolated and then transferred back to the uids grid. The solution was considered converged when the four variables, and velocities, charge, and potential, fell within set tolerances at all grid points in successive iterations. If a converged solution did not satisfy the assigned electrostatic boundary condition at the three gas phase boundaries, the domain was enlarged in the appropriate direction and the procedure repeated. The computations were performed on a VAX 4000 using a FORTRAN CODE. A typical computation involved approximately 10 nodes and required about 10 min CPU time, indicating a high efciency of the numerical algorithm. The algorithm was veried against the well-known Blasius problem [20]. V. RESULTS DISCUSSION

Fig. 2. Corona current versus potential difference of electrodes for 100-m wires at various gap spacing.

AND

Two case studies were evaluated, each one with and without ion deposition/removal permitted. In case study I, the positive (upstream) and negative (downstream) electrodes were biased and V, respectively, with the at remaining surfaces grounded. In case study II, the positive and and negative electrodes were biased at V, respectively, with the remaining surfaces at the potentials of the corresponding electrodes. The diameter of the electrodes was 2 m for a gap of 25 mm. The distance of the positive electrode downstream from the leading edge was 25 or 100 mm. The computational domain was approximately 12 gap lengths in height and nine gaps in length. Fig. 2 is an example of the voltagecurrent ( ) characteristic for 100- m wires at different gap lengths using (7)(9). The current is per unit length of wire. As expected, a decrease in the gap size leads to an increase in the discharge current at a constant potential. A comparison of Figs. 3 and 4 for case study I shows a nonlinearity in surface conduction with ion deposition/removal permitted. The nonlinear potential is possible through (13), even though the surface itself is ohmic. A large potential drop also exists near each electrode in Fig. 3. In case study I, positive ions produced at the positive electrode accept electrons at either the negative electrode or along the zero potential surface, giving the asymmetrical potential distribution shown in Fig. 3. In case study II, the bias voltage is unsymmetrical, as shown in Fig. 5. Case study II, with ion deposition/removal (Fig. 5), again shows a nonlinear potential drop along the ohmic surface. As will be discussed, a free-stream velocity

Fig. 3. Case study I, potential distribution with tion/removal (free-stream velocity 0).

=

x;

with ion deposi-

of 15 m/s (Reynolds number 5 10 ) is shown to have a negligible effect on the potential distribution. A dropoff in potential is observed near each electrode as in case study I. Fig. 6 is a larger scale plot of the potential for case study II demonstrating several features, including a linear potential drop between the electrodes (without ion deposition), constant potential surfaces upstream and downstream of the electrodes, and a gradual dropoff in potential upstream of the leading edge by (16). At increased distances from the surface, the eld lines are directed either upstream or downstream from the positive electrode, as shown in Fig. 7. A corresponding dropoff in eld magnitude is not apparent here because of the normalized plots. In Fig. 8, an upstream direction of the electric eld is not indicated by the limited size of the plot. However, the direction of eld lines can be inferred from Fig. 6 together with (4).



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Fig. 4. Case study I, potential distribution with tion/removal (free-stream velocity 0).

=

x;

without ion deposi-

Fig. 6. Case study II, potential distribution with tion/removal (free-stream velocity 0).

=

x;

without deposi-

Fig. 5. Case study II, potential distribution with tion/removal (free-stream velocity 15 m/s).

=

x;

with ion deposi-

Fig. 7. Case study I, normalized vector eld plot of electric eld strength E with x; without ion deposition/removal (free-stream velocity 0).

=

A close examination of the vector eld plot in Fig. 8 reveals that ion current is being removed at the surface near the positive electrode and deposited near the downstream (negative electrode). A similar effect can be seen in Fig. 9 for the current density . The effect does not appear in Fig. 7, since ion deposition and removal are not permitted. The eld lines in Fig. 8 enter normal to the constant potential surfaces both upstream and downstream of the electrodes. The positive ion charge density distributions are shown in Figs. 1012. Of interest is the comparison between Figs. 10 and 11 for case study I, in which the charge density is seen to be reduced with ion deposition/removal permitted. Considering the reduction in charge to be the result of a simple short circuit of the current path from the gas to the ohmic surface does not adequately account for the signicant plate drag reduction observed in [18]. An alternative explanation with ion deposition/removal is that the shift in charge distribution is the result of the decreased potential as a boundary condition on the surface, as can be seen by a comparison of Figs. 3 and 4.

In Fig. 12, a peak in the charge distribution results for the unsymmetrical voltage bias of case study II (see Fig. 5). In the absence of ion deposition/removal (not shown), the peaks of Fig. 12 are shifted left toward the positive electrode and exhibit an increase in magnitude of the charge density as noted for case I above [24]. Fig. 13 shows the velocity proles with and without corona discharge for case study II (with ion deposition). With corona, an increase in the velocity through the boundary layer is apparent with a corresponding increase in shear at the wall. However, a reduction in the net plate drag results when the force on the electrodes is taken into account. A decrease in the local shear was also observed when ion deposition was allowed, probably as a result of a decrease in the ion concentration in the boundary layer [18]. The effect of convection on current becomes important when the free-stream velocity inuences the eld drift velocity of ions. This is demonstrated in Fig. 14, where the magnitude is plotted for case II at a constant of the current density height of 8.2 mm between the electrodes with the free-stream


392


Fig. 8. Case study II, normalized vector eld plot of electric eld strength E with x; with ion deposition/removal (free-stream velocity 0).

=

Fig. 11. Case study I, charge density distribution at different levels above surface with x; without ion deposition (free-stream velocity 0).

=

Fig. 9. Case study I, normalized vector eld plot of current density x; with ion deposition/removal (free-stream velocity 0).

=

J with

Fig. 12. Case study II, charge density distribution at different levels above surface; with ion deposition (free-stream velocity 0).

=

Fig. 10. Case study I, charge density distribution at different levels above surface with x; with ion deposition (free-stream velocity 0).

=

velocity at 0 and 15 m/s. The magnitude of peaks between the electrodes as would be expected from (12) and Fig. 12 for the charge density distribution. An examination of the numerical data used in Fig. 14 showed that mainly the component of the current density was affected by the freecomponent of remained stream convection, while the largely unchanged. One can utilize (12) to nd a value for the convection . velocity that will inuence the ion drift velocity Taking a nominal value of the electric eld strength V/0.025 m = V/m from Fig. 4 at , an ion charge C/m (peak current density in Fig. 10, density ), and a positive ion mobility m /V s m/s and A/m , showing that this gives estimated current density compares favorably with the more accurately computed peak current density in Fig. 14 at zero ). A conservative estimate is that free-stream velocity ( the zero free-stream velocity gures in this paper (Figs. 3 and 4, etc.) can be considered accurate up to velocities of about 5 m/s ( 35 m/s).



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ACKNOWLEDGMENT The authors wish to thank Prof. J. M. Prusa, Iowa State University, for his remarks on the numerical aspects of this study and Prof. J. F. Hoburg, Carnegie Mellon University, for his input on the electrostatics. REFERENCES[1] J. Cobine, Gaseous Conduction. New York: Dover, 1958. [2] L. B. Leob, Fundamental Processes of Electrical Discharge in Gases. New York: Wiley, 1939. [3] J. Cross, Electrostatics: Principles, Problems and Applications. Bristol, U.K.: Adam Hilger, 1986. [4] J. L. Davis and J. F. Hoburg, Wire-duct precipitator eld and charge computation using nite element and characteristics methods, J. Electrostatics, vol. 14, pp. 187199, Aug. 1983. [5] H. R. Velkoff and R. Godfrey, Low-velocity heat transfer to a at plate in the presence of a corona discharge in air, J. Heat Transfer, vol. 101, pp. 157163, Feb. 1979. [6] H. R. Velkoff, Evaluating the interactions of electrostatic elds with uid ows, presented at the ASME Design Eng. Conf., New York, NY, Apr. 1922, 1971, Paper 71-DE-41. [7] H. R. Velkoff and F. A. Kulacki, Electrostatic cooling, presented at the ASME Design Eng. Conf., Chicago, IL, May 912, 1977, Paper 77-DE-36. [8] K. G. Kibler and H. G. Carter, Jr., Electrocooling in gases, J. Appl. Phys., vol. 45, pp. 44364440, 1974. [9] H. J. White, Industrial Electrostatic Precipitation. Reading, MA: Addison-Wesley, 1963. [10] J. Lawton and F. J. Weinberg, Electrical Aspects of Combustion. Oxford, U.K.: Clarendon, 1969. [11] G. M. Colver and Y. Nakai, Flame stabilization by an electrical discharge and ame visualization of the inuence of a semi-insulating wall on the ionic wind, in Proc. 13th IEEE-IAS Annu. Meeting, Toronto, Ont., Canada, Oct. 1978, pp. 95104. [12] D. M. Bushnell, Turbulent drag reduction for external ows, presented at the AIAA 21st Aerospace Sciences Meeting, Reno, NV, Jan. 1983, Paper AIAA-83-0227. [13] M. R. Malik, L. M. Weinstein, and M. Y. Hussaini, Ion wind drag reduction, presented at the AIAA 21st Aerospace Sciences Meeting, Reno, NV, Jan. 1983, Paper AIAA-83-0231. [14] V. F. Medvedev and M. S. Krakov, Flow separation by means of magnetic uid, J. Magnetism Magn. Mater., vol. 39, no. 1/2, pp. 119122, Nov. 1983. [15] D. A. Nelson, M. M. Ohdi, R. L. Whipple, S. Zia, and A. Ansari, Effects of corona discharge on forced convection in tubes: Pressure drop results, in Conf. Rec. IEEE-IAS Annu. Meeting, San Diego, CA, Oct. 1989, pp. 20902093. [16] F. Soetomo, The inuence of high voltage discharge on at plate drag at low Reynolds number air ow, M.S. thesis, Dep. Mech. Eng., Iowa State Univ., Ames, 1992. [17] L. Chaoyu and J. Roth, Boundary layer control by a one atmosphere uniform discharge plasma, in Conf. Rec. IEEE Conf. Plasma Science, Madison, WI, June 1995, p. 51. [18] S. El-Khabiry and G. M. Colver, Drag reduction by dc corona discharge along an electrically conductive at plate for small Reynolds number ow, Phys. Fluids, vol. 9, no. 3, pp. 587599, Mar. 1997. [19] S. El-Khabiry and G. M. Colver, A semi-theoretical development of drag reduction on a thin plate in low Reynolds number ow with corona discharge, in Proc. Electrostatics Society of America Annu. Meeting, University of Rochester, Rochester, NY, June 2023, 1995, pp. 135138. [20] H. H. Schlichting, Boundary Layer Theory, 4th ed. New York: McGraw-Hill, 1960. [21] G. M. Colver, F. Hossain, F. Soetomo, D. Tucholski, and J. Wang, Bubble and elutriation control in uidized beds with electric elds (including summary reports: Electrostatic separation of powder in a circulating bed; corona discharge), in 1992 IFPRI Annu. Rep., International Fine Particle Research Institute, Widnes, Cheshire, U.K., Dec. 31, 1992. [22] A. E. Seaver, Onset potential for unipolar charge injection, in Conf. Rec. IEEE-IAS Annu. Meeting, Toronto, Ont., Canada, Oct. 1993, pp. 17. [23] S. Oglesby and G. Nichols, A manual of electrostatic precipitator technologyPart I: Fundamentals, Southern Research Inst., Birmingham, AL, Aug. 1970. [24] S. H. El-Khabiry, Numerical evaluation of corona discharge as a means of boundary layer control and drag reduction, Ph.D. dissertation, Dep. Mech. Eng., Iowa State Univ., Ames, 1994.

Fig. 13. Case study II, velocity prole 13 mm downstream from positive 3 electrode with/without corona; with ion deposition (free-stream velocity m/s, Reynolds number 7.6 103 ).

=

2

=

Fig. 14. Case study II, current density (magnitude of J ) distribution with position x at 8.2 mm above surface; with ion deposition (free-stream velocity 0 and 15 m/s).

=

VI. SUMMARY This paper does not attempt to validate the mechanisms of ion deposition or removal, but rather gives predictive results should these mechanisms occur. In a separate study [18], the trend of decreasing drag force on a conductive plate with an increase in corona voltage predicted with the present theory is found to be consistent with the experimental results of Soetomo [16]. Some additional observations from this study are as follows. The currentvoltage characteristic gives acceptable agreement with expected trends of corona discharge along a semi-insulating surface. The charge, current, and potential distributions of the corona discharge are dependent on the voltage bias of the electrodes. Boundary layer convection velocities less than about 5 m/s have a small effect on the charge, current, and potential distributions of the discharge.


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[25] W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed. New York: Wiley, 1976, ch. 17. [26] H. L. Curtis, Insulating properties of solid dielectrics, Bureau of Standards Bull., U.S. Dep. Commerce, Washington, DC, vol. 11, pp. 359420, 1915. [27] J. P. Lacharme, Ionic jump processes and high eld conduction in glasses, J. Non-Cryst. Solids, vol. 27, no. 3, pp. 381397, Mar. 1978. [28] D. C. Jolly and S. T. Chu, Surface electrical breakdown of tin oxide coated glass, J. Appl. Phys., vol. 50, no. 10, pp. 61966199, 1979. [29] H. T. M. Haenen, The characteristic decay with time of surface charges on dielectrics, J. Electrostatics, vol. 1, no. 2, pp. 173185, May 1975. [30] B. Moslehi, Electromechanics and electrical breakdown of particulate layers, High Temperature Gasdynamics Lab., Stanford Univ., Stanford, CA, Rep. T-236, Dec. 1983. [31] T. G. Beuthe and J. S. Chang, Gas discharge phenomena, in Handbook of Electrostatic Processes, J. S. Chang, A. J. Kelly, and J. M. Crowley, Eds. New York: Marcel Dekker, 1995, ch. 9. [32] H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. [33] D. A. Anderson, J. C. Tannehill, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer. New York: Hemisphere, 1984. [34] S. V. Patankar and D. B. Spalding, Heat and Mass Transfer in Boundary Layer. London, U.K.: Morgan-Grampian, 1967.

Samir El-Khabiry received the B.Sc. degree, as a distinguished student, from the Aeronautical Engineering Department, Military Technical College, Cairo, Egypt, in 1974, the M.Sc. degree from Cairo University, Cairo, Egypt, in 1983, and the M.Eng. degree in aerospace engineering and engineering mechanics and the Ph.D. degree in mechanical engineering from Iowa State University, Ames, in 1991 and 1994, respectively. He was a Temporary Assistant Professor in the Mathematics Department, Iowa State University, from Fall 1995 to June 1998. He is currently a Research and Development Engineer with Bergstrom Inc., Rockford, IL. He is also a Special Advisor for Distance Education for the International Institute of Theoretical and Applied Physics, Ames, IA. His interests include environmental applications of uid mechanics and climate control, including computational and experimental uid dynamics, thermodynamics and heat transfer with applications in optimization, two-phase ow, and atmospheric boundary layers and boundary layer control. He is also developing interactive calculus courses for the Internet.

Gerald M. Colver received the B.S.M.E degree from Bradley University, Peoria, IL, in 1962 and the M.S.M.E. and Ph.D. degrees from the University of Illinois, Urbana, in 1964 and 1969, respectively. During 19691970, he was a Postdoctoral Researcher at Imperial College, London, U.K., and from 1970 to 1977, he was an Assistant Professor of Mechanical Engineering at Rensselaer Polytechnic Institute, Troy, NY. In September 1977, he joined the faculty of Iowa State University, Ames, where he is currently a Professor in the Department of Mechanical Engineering. He has lectured widely in the U.S. and abroad and has published in the areas of multiphase ow and combustion. He is also an Industrial Consultant on powder technology and has given an industrial short course on the topic of electrostatic effects in powders for the Fine Particle Society. He has recently authored a review chapter on electrostatic instrumentation in multiphase ow. Prof. Colver is a Registered Professional Engineer in the State of New York.


modelling of corona discharge

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